Problem: Solve for $x$ and $y$ by deriving an expression for $y$ from the second equation, and substituting it back into the first equation. $\begin{align*}4x-3y &= 6 \\ -5x+y &= 1\end{align*}$
Solution: Begin by moving the $x$ -term in the second equation to the right side of the equation. $y = {5x+1}$ Substitute this expression for $y$ in the first equation. $4x-3({5x + 1}) = 6$ $4x - 15x - 3 = 6$ Simplify by combining terms, then solve for $x$ $-11x - 3 = 6$ $-11x = 9$ $x = -\dfrac{9}{11}$ Substitute $-\dfrac{9}{11}$ for $x$ back into the top equation. $4( -\dfrac{9}{11})-3y = 6$ $-\dfrac{36}{11}-3y = 6$ $-3y = \dfrac{102}{11}$ $y = -\dfrac{34}{11}$ The solution is $\enspace x = -\dfrac{9}{11}, \enspace y = -\dfrac{34}{11}$.